Algebra Examples

$5 - 7^{x + 1} = 3$

Step 1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.1

Subtract $5$ from both sides of the equation.

$- 7^{x + 1} = 3 - 5$

Step 1.2

Subtract $5$ from $3$ .

$- 7^{x + 1} = - 2$

Step 2

Divide each term in $- 7^{x + 1} = - 2$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- 7^{x + 1} = - 2$ by $- 1$ .

$\frac{- 7^{x + 1}}{- 1} = \frac{- 2}{- 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{7^{x + 1}}{1} = \frac{- 2}{- 1}$

Step 2.2.2

Divide $7^{x + 1}$ by $1$ .

$7^{x + 1} = \frac{- 2}{- 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $- 2$ by $- 1$ .

$7^{x + 1} = 2$

Step 3

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (7^{x + 1}) = \ln (2)$

Step 4

Expand $\ln (7^{x + 1})$ by moving $x + 1$ outside the logarithm.

$(x + 1) \ln (7) = \ln (2)$

Step 5

Simplify the left side.

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Step 5.1

Simplify $(x + 1) \ln (7)$ .

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Step 5.1.1

Apply the distributive property.

$x \ln (7) + 1 \ln (7) = \ln (2)$

Step 5.1.2

Multiply $\ln (7)$ by $1$ .

$x \ln (7) + \ln (7) = \ln (2)$

Step 6

Move all the terms containing a logarithm to the left side of the equation.

$x \ln (7) + \ln (7) - \ln (2) = 0$

Step 7

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$x \ln (7) + \ln (\frac{7}{2}) = 0$

Step 8

Subtract $\ln (\frac{7}{2})$ from both sides of the equation.

$x \ln (7) = - \ln (\frac{7}{2})$

Step 9

Divide each term in $x \ln (7) = - \ln (\frac{7}{2})$ by $\ln (7)$ and simplify.

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Step 9.1

Divide each term in $x \ln (7) = - \ln (\frac{7}{2})$ by $\ln (7)$ .

$\frac{x \ln (7)}{\ln (7)} = \frac{- \ln (\frac{7}{2})}{\ln (7)}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $\ln (7)$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{x \ln (7)}{\ln (7)} = \frac{- \ln (\frac{7}{2})}{\ln (7)}$

Step 9.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (\frac{7}{2})}{\ln (7)}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{7}{2})}{\ln (7)}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (\frac{7}{2})}{\ln (7)}$

Decimal Form:

$x = - 0.64379281 \dots$